On Linux you can use strace to see which system calls a program uses. The following will list all "open(2)" system calls and filter them through sed to show, hopefully, the terminfo file used by tput to translate the terminfo capability for the current terminal.

Note that that sed expression assumes that your terminfo files are stored in a path with the pattern "terminfo" in it. If that worries you then you could use a different assumption and pick the last "open" system call.